• cycle with a unique chord;
  • decomposition;
  • structure;
  • detection;
  • recognition;
  • Heawood graph;
  • Petersen graph;
  • coloring


We give a structural description of the class �� of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in �� is either in some simple basic class or has a decomposition. Basic classes are chordless cycles, cliques, bipartite graphs with one side containing only nodes of degree 2 and induced subgraphs of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for ��, i.e. every graph in �� can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations, and all graphs built this way are in ��. This has several consequences: an ��(nm) -time algorithm to decide whether a graph is in ��, an ��(n+ m) -time algorithm that finds a maximum clique of any graph in ��, and an ��(nm) -time coloring algorithm for graphs in ��. We prove that every graph in �� is either 3-colorable or has a coloring with ω colors where ω is the size of a largest clique. The problem of finding a maximum stable set for a graph in �� is known to be NP-hard. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 31–67, 2010